Calc

What is the washer method?

V = π ∫(R² − r²) dx — finds volume of a solid of revolution by subtracting the inner radius squared from the outer radius squared to account for the hollow region

When can you use Rolle’s Theorem?

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f′(c)=0 — guarantees at least one point where the tangent line is horizontal

What are Riemann sums?

Σ f(xᵢ)Δx where Δx = (b−a)/n — approximates area under a curve using rectangles by adding height times width, with left/right/midpoint choices affecting accuracy

Δx = (b − a)/n, then multiplying each rectangle’s height f(Mᵢ) (a sample point in the i-th subinterval) by Δx and summing: Lim as n approaches infinity Σ f(Mᵢ)Δx from i = 1 to n.

What is a normal line?

slope = −1/(slope of tangent) — a line perpendicular to the tangent line at a point on a curve

What is the limit definition of a derivative?

lim h→0 [f(x+h) − f(x)]/h — defines the derivative as the instantaneous rate of change and slope of the tangent line

What is the average value of a function?

(1/(b−a)) ∫ from a to b f(x) dx — gives the average height of a function over an interval using the definite integral

What does the Mean Value Theorem say?

f′(c) = (f(b) − f(a)) / (b − a) — guarantees a point where instantaneous rate of change equals average rate of change on [a,b]

continuous [a,b]

Differentiable (a,b)

What is average rate of change?

(f(b) − f(a)) / (b − a) — measures the slope of the secant line between two points on a function

How are position, velocity, and acceleration related?

velocity = f′(x), acceleration = f″(x) — velocity is the derivative of position and acceleration is the derivative of velocity

What is linear approximation?

y − y₁ = m(x − x₁) — uses the tangent line at a point to estimate nearby values of a function

What is a point of inflection?

f″(x)=0 and changes sign — a point where concavity changes from up to down or down to up

Difference between total distance and displacement?

distance = ∫ from a to b |v(t)| dt, displacement = ∫ from a to b v(t) dt — distance measures total movement regardless of direction, displacement measures net change in position

What does the Fundamental Theorem of Calculus Part 1 do?

∫ from a to b f(x) dx = F(b) − F(a) — evaluates definite integrals using antiderivatives by subtracting values at endpoints

What happens when you differentiate an integral? Second fundamental theorem of calc

d/dx ∫ from x to a f(t) dt = f(x) — differentiation cancels integration, returning the original function (use chain rule if limits depend on x)

What is a slope field?

no single formula — a visual representation of a differential equation showing slopes of solution curves at many points

What is instantaneous rate of change?

f′(x) — the derivative representing the slope of the tangent line at a specific point

When do you use L’Hôpital’s Rule?

lim as x approaches a f(x)/g(x) = lim as x approaches a f′(x)/g′(x) — used to evaluate indeterminate limits like 0/0 or ∞/∞ by differentiating numerator and denominator

What is the trapezoidal rule?

(Δx/2)[f(x₀)+2f(x₁)+2f(x2)+…+2f(xₙ-1)]

Delta x = (b-a)/n

— approximates area under a curve using trapezoids for a more accurate estimate than rectangles

What are critical points?

f′(x)=0 or undefined — points where a function may have local maxima or minima and must be tested using derivatives and endpoints

Chain rule

d/dx [f(g(x))] = f’(g(x)) · g’(x)